cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A099456 Expansion of 1/(1 - 4*x + 5*x^2).

Original entry on oeis.org

1, 4, 11, 24, 41, 44, -29, -336, -1199, -3116, -6469, -10296, -8839, 16124, 108691, 354144, 873121, 1721764, 2521451, 1476984, -6699319, -34182196, -103232189, -242017776, -451910159, -597551756, -130656229, 2465133864
Offset: 0

Views

Author

Paul Barry, Oct 16 2004

Keywords

Comments

Associated to the knot 9_44 by the modified Chebyshev transform A(x) -> (1/(1+x^2)^2)A(x/(1+x^2)). See A099457 and A099458.
Imaginary part of (2+i)^n. - Gary W. Adamson, Apr 05 2008; Franklin T. Adams-Watters, Jan 06 2009

Links

Crossrefs

Cf. A139011, A090131 (inv. bin. transf.)

Programs

  • Maple
    seq(((2+I)^(n+1) - (2-I)^(n+1))/(2*I),n=0..30);  # James R. Buddenhagen, Dec 29 2017
  • Mathematica
    CoefficientList[Series[1/(1-4*x+5*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 09 2013 *)
Table[((1+2*I)*(2-I)^n + (1-2*I)*(2+I)^n)/2,{n,0,20}] (* Vaclav Kotesovec, Oct 09 2013 *)
  • Sage
    [lucas_number1(n,4,5) for n in range(1, 29)] # Zerinvary Lajos, Apr 22 2009

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-5)^k*4^(n-2k).
E.g.f. (with offset 1): exp(2*x)*sin(x). - Zerinvary Lajos, Apr 06 2009 [corrected by Joerg Arndt, Apr 24 2011]
a(n) = 4*a(n-1) - 5*a(n-2), a(0)=1, a(1)=4. - Vincenzo Librandi, Mar 22 2011
From Paul Curtz, Apr 24 2011: (Start)
a(n) - a(n-4) = 40 * A118444(n);
a(n) - a(n-2) = 10 * A139011(n). (End)
a(n) = ((1+2*i)*(2-i)^n + (1-2*i)*(2+i)^n)/2. - Vaclav Kotesovec, Oct 09 2013
a(n) = ((2+i)^(n+1) - (2-i)^(n+1))/(2*i).
Lim sup n->infinity |a(n)|/5^((n+1)/2) = 1. - Vaclav Kotesovec, Oct 09 2013
a(n) = Sum_{k=0..n} (-1)^k*2^(n-2*k)*binomial(n+1,2*k+1). - Gerry Martens, Sep 18 2022
E.g.f.: exp(2*x)*(cos(x) + 2*sin(x)). - Stefano Spezia, Jul 24 2025

A139011 Real part of (2 + i)^n, where i = sqrt(-1).

Original entry on oeis.org

1, 2, 3, 2, -7, -38, -117, -278, -527, -718, -237, 2642, 11753, 33802, 76443, 136762, 164833, -24478, -922077, -3565918, -9653287, -20783558, -34867797, -35553398, 32125393, 306268562, 1064447283, 2726446322, 5583548873, 8701963882
Offset: 0

Views

Author

Gary W. Adamson, Apr 05 2008

Keywords

Comments

Imaginary part of (2 + i)^n gives A099456.
Irrespective of signs, odd-indexed terms of A006496 interleaved with even-indexed signs of A006495.
Binomial transform of A146559, second binomial transform of A056594. - Philippe Deléham, Dec 02 2008

Examples

			1 + 2*x + 3*x^2 + 2*x^3 - 7*x^4 - 38*x^5 - 117*x^6 - 278*x^7 - 527*x^8 + ...
a(5) = -38 since (2 + i)^5 = (-38 + 41*i).
a(5) = -38 since [2,-1; 1,2]^5 = [ -38,-41; 41,-38], where 41 = A099456(5).
a(5) = -38 = A006496(5).

Links

Crossrefs

Cf. A099456, A006495, A006496, A056594, A146559 (inv bin. transf.).

Programs

  • Magma
    [ Integers()!Real((2+Sqrt(-1))^n): n in [0..29] ];  // Bruno Berselli, Apr 26 2011
  • Maple
    restart: G(x):=exp(x)^2*cos(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=1..29 ); # Zerinvary Lajos, Apr 06 2009
  • Mathematica
    Re[(2+I)^Range[0,30]] (* or *) LinearRecurrence[{4,-5},{1,2},30] (* Harvey P. Dale, Nov 02 2022 *)
  • PARI
    a(n) = real((2 + I)^n) /* Michael Somos, Dec 26 2009 */
  • PARI
    Vec((1 - 2*x) / (1 - 4*x + 5*x^2) + O(x^30)) \\ Colin Barker, Sep 22 2017
  • Sage
    [lucas_number2(n,4,5)/2 for n in range(0,31)] # Zerinvary Lajos, Jul 08 2008

Formula

Real part of (2 + i)^n, i^2 = -1.
Term (1,1) of matrix [2,-1; 1,2]^n.
(a(n))^2 + (A099456(n))^2 = 5^n.
From R. J. Mathar, Apr 06 2008: (Start)
O.g.f.: (1-2x) /(1-4x+5x^2).
a(n) = 4*a(n-1) - 5*a(n-2) = 2*A099456(n-1) - 5*A099456(n-2). (End)
E.g.f.: exp(x)^2*cos(x). - Zerinvary Lajos, Apr 06 2009
a(-n) = a(n) / 5^n. - Michael Somos, Dec 26 2010
a(n) = Sum_{k=0..n} A098158(n,k)*2^(2k-n)*(-1)^(n-k). - Philippe Deléham, Dec 02 2008
2*a(n) - a(n+1) = A099456(n-1) for n>0. First differences are (up to sign) A118444. - Paul Curtz, Apr 25 2011
a(n) = Sum_{k=0..n} A201730(n,k)*(-2)^k. - Philippe Deléham, Dec 06 2011
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*2^(n-2*k)*binomial(n,2*k). - Gerry Martens, Sep 18 2022

Extensions

Cross-reference corrected by Franklin T. Adams-Watters, Jan 06 2009
Added a(0)=1 by Michael Somos, Dec 26 2010
Edited by Franklin T. Adams-Watters, Apr 10 2011

A118443 Row sums of triangle A118441, which is the matrix log of triangle A118435.

Original entry on oeis.org

1, -2, 3, -36, -155, 474, 1127, -1992, -1719, -4810, -31669, 109332, 286637, -596974, -904785, 449136, -3218287, 16156782, 50232979, -121747380, -233735691, 309853258, 15768823, 1624290984, 6853579225, -19712646746, -44873974053, 79998871428, 90434035261
Offset: 0

Views

Author

Paul D. Hanna, Apr 28 2006

Keywords

Links

Crossrefs

Cf. A118441 (triangle), A118442 (column 0), A118444 (a(n)/(n+1)); A118435.

Programs

  • Mathematica
    nmax = 30;
h[n_, k_] := Binomial[n, k]*(-1)^(Quotient[n+1, 2] - Quotient[k, 2]+n-k);
H = Table[h[n, k], {n, 0, nmax}, {k, 0, nmax}];
Cn = Table[Binomial[n, k], {n, 0, nmax}, {k, 0, nmax}];
L = MatrixLog[H.Inverse[Cn].H];
Total /@ Rest@L (* Jean-François Alcover, Apr 08 2024 *)
  • PARI
    {a(n)=polcoeff((1+x)*(1-3*x+18*x^2-78*x^3+45*x^4-175*x^5)/(1+6*x^2+25*x^4 +x*O(x^n))^2,n)}

Formula

G.f.: (1+x)*(1-3*x+18*x^2-78*x^3+45*x^4-175*x^5)/(1+6*x^2+25*x^4)^2.
E.g.f.: cos(2*x)*((1 - x)*cosh(x) + (1 + 3*x)*sinh(x)) - sin(2*x)*((1 + x)*cosh(x) - (1 - 3*x)*sinh(x)). - Stefano Spezia, Jul 01 2023
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.